Sub-Lorentzian geometry on anti-de Sitter space

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• 作者：Der-Chen Chang ; Irina Markina ; Alex ; er Vasil'ev
• 关键词：Sub-Riemannian and sub-Lorentzian geometries ; Geodesic ; Anti-de Sitter space
• 刊名：Journal des Mathematiques Pures et Appliquees
• 出版年：2008
• 出版时间：July 2008
• 年：2008
• 卷：90
• 期：1
• 页码：82-110
• 全文大小：292 K

文摘

Sub-Riemannian Geometry is proved to play an important role in many applications, e.g., Mathematical Physics and Control Theory. Sub-Riemannian Geometry enjoys major differences from the Riemannian being a generalization of the latter at the same time, e.g., geodesics are not unique and may be singular, the Hausdorff dimension is larger than the manifold topological dimension. There exists a large amount of literature developing sub-Riemannian Geometry. However, very few is known about its extension to pseudo-Riemannian analogues. It is natural to begin such a study with some low-dimensional manifolds. Based on ideas from sub-Riemannian geometry we develop sub-Lorentzian geometry over the classical 3-D anti-de Sitter space. Two different distributions of the tangent bundle of anti-de Sitter space yield two different geometries: sub-Lorentzian and sub-Riemannian. We use Lagrangian and Hamiltonian formalisms for both sub-Lorentzian and sub-Riemannian geometries to find geodesics.